Fourier transform for a signal to be transmitted on a random access channel

ABSTRACT

Method and apparatus for processing a signal using a recursive method for determining a plurality of frequency components of the signal, the signal being a chirp-like polyphase sequence, wherein a first frequency component of the plurality of frequency components is determined; a component factor is determined by accessing a factor table for use in determining a second frequency component of the plurality of frequency components; and the second frequency component is determined using the determined first frequency component and the determined component factor.

FIELD OF THE INVENTION

This invention relates to processing a signal to be transmitted on arandom access channel, in particular where the signal is for use as arandom access channel preamble.

BACKGROUND

In a wireless cellular communication system, the procedure ofestablishing communication between a mobile terminal or User Equipment(UE) and a base station is called random access. Such random access canbe implemented using a random access channel (RACH) in, for example, anorthogonal frequency division multiplexing (OFDM) communication systemor a single carrier frequency division multiplexing (SC-FDMA)communication system. Random access enables the establishment of theuplink from a UE to a base station. Using the RACH, a UE can send anotification to the network indicating that the UE has data to transmit.Receipt of the notification at the base station allows the base stationto estimate the UE timing, to thereby realize uplink synchronizationbetween the UE and the base station.

The random access channel (RACH) typically consists of a ranging signalor a preamble. The preamble is designed to allow the base station todetect the random access attempt within target detection and false alarmprobabilities, and to minimise the impact of collisions on the RACH, asis known in the art. Moreover, the base station should be able to detectseveral simultaneous random preambles sent from different UEs andcorrectly estimate the timing of each of the UEs. In order to achievethat goal, the RACH preambles should have i) good cross-correlationproperties to allow for accurate timing estimation of differentsimultaneous and asynchronous RACH preambles, ii) good auto-correlationproperties to allow for accurate timing estimation, iii) zerocross-correlation for synchronous and simultaneous RACH preambles.

Long Term Evolution (LTE) wireless networks, also known as EvolvedUniversal Terrestrial Radio Access Networks (E-UTRAN), are beingstandardized by the 3GPP working groups. The Orthogonal FrequencyDivision Multiple Access (OFDMA) access scheme and the Single CarrierFrequency Division Multiple Access (SC-FDMA) access scheme were chosenfor the downlink (DL) and the uplink (UL) of E-UTRAN, respectively.Signals from different User Equipments (UEs) to a base station are timeand frequency multiplexed on a physical uplink shared channel (PUSCH).In the case that the UE is not UL synchronized, the UE uses anon-synchronized Physical Random Access Channel (PRACH) to communicatewith the base station, and in response the base station provides ULresources and timing advance information to allow the UE to transmit onthe PUSCH.

The 3GPP RAN Working Group 1 (WG1) has agreed on the preamble basedphysical structure of the PRACH (as described in “3GPP TS 36.211 EvolvedUniversal Terrestrial Radio Access (E-UTRA); Physical channels andmodulation”). The RAN WG1 also agreed on the number of availablepreambles that can be used concurrently to minimize the collisionprobability between UEs accessing the PRACH in a contention-basedmanner. The Zadoff-Chu (ZC) sequence has been selected for RACHpreambles for LTE networks.

A Zadoff-Chu sequence is a complex-valued mathematical sequence which,when used for radio signals, gives rise to a signal, whereby cyclicallyshifted versions of the signal do not cross-correlate with each otherwhen the signal is recovered, for example at the base station. Agenerated Zadoff-Chu sequence that has not been shifted is known as a“root sequence”. The Zadoff-Chu sequence exhibits the useful propertythat cyclically shifted versions of the sequence remain orthogonal toone another, provided that each cyclic shift, when viewed within thetime domain of the signal, is greater than the combined propagationdelay and multi-path delay-spread of the signal as it is transmittedbetween the UE and base station.

The complex value at each position (n) of each root (μ) of theZadoff-Chu sequence (for odd N_(ZC), where N_(ZC) is the length of theZadoff-Chu sequence) is given by:

${{x_{\mu}(n)} = ^{{- j}\; \frac{\pi \; \mu \; {n{({n + 1})}}}{N_{ZC}}}},$

where 0≦n≦N_(ZC)−1.

All of the RACH preambles are generated by cyclic shifts of a number ofroot sequences of the Zadoff-Chu sequence, which are configurable on acell-basis. A RACH preamble is transmitted from a UE to the base stationto allow the base station to estimate, and if needed, adjust the timingof the UE transmission. It has been agreed by the RAN WG1 that there area total of 64 RACH preambles allocated for each cell of a base station.Specifically, a cell can use different cyclically shifted versions ofthe same ZC root sequence, or other ZC root sequences if needed, as RACHpreambles. To maximize the number of available Zadoff-Chu sequences fora certain sequence length (N_(ZC)) it is preferred to choose thesequence length as a prime number, and therefore an odd number.Typically for LTE, the length of the Zadoff-Chu may be for example 839or 139 depending on the format of the RACH preamble.

For the uplink in LTE wireless networks, SC-FDMA is used which is asingle-carrier transmission based on Discrete Fourier Transform (DFT)spread OFDM. With reference to FIG. 1, the principle of SC-FDMA will nowbe described. FIG. 1 shows a user equipment 101 and a base station 121.The user equipment comprises a serial to parallel block 102 with aserial input for receiving a serial signal. The user equipment 101further comprises an N-point Discrete Fourier Transform (DFT) block 104,a subcarrier mapping block 106, an M-point Inverse Discrete FourierTransform (IDFT) block 108, a parallel to serial block 110, a CyclicPrefixing (CP) and Pulse shaping (PS) block 112, a Digital to Analogueconverter (DAC) and Radio Frequency (RF) converter block 114 and anantenna 116 for transmitting a signal over a channel 118 of thecommunication system. The serial to parallel block 102 has a paralleloutput coupled to a parallel input of the N-point DFT block 104. Aparallel output of the N-point DFT block 104 is coupled to a parallelinput of the subcarrier mapping block 106. A parallel output of thesubcarrier mapping block 106 is coupled to a parallel input of theM-point IDFT block 108. A parallel output of the M-point IDFT block 108is coupled to a parallel input of the parallel to serial block 110. Aserial output of the parallel to serial block 110 is coupled to an inputof the CP and PS block 112. An output of the CP and PS block 112 iscoupled to an input of the DAC and RF converter block 114. An output ofthe DAC and RF converter block 114 is coupled to the antenna 116.

The base station 121 comprises an antenna 120 for receiving a signalover the channel 118 of the communication system. The base stationfurther comprises a Radio Frequency (RF) converter and Analogue toDigital converter (ADC) block 122, a remove Cyclic Prefixing (CP) block124, a serial to parallel block 126, an M-point Discrete FourierTransform (DFT) block 128, a subcarrier demapping and equalization block130, an N-point Inverse Discrete Fourier Transform (IDFT) block 132, aparallel to serial block 134 and a detection block 136 for detecting thesignals.

A serial output of the antenna 120 is coupled to a serial input of theRF converter and ADC block 122. A serial output of the RF converter andADC block 122 is coupled to a serial input of the remove CP block 124. Aserial output of the remove CP block 124 is coupled to a serial input ofthe serial to parallel block 126. A parallel output of the serial toparallel block is coupled to a parallel input of the M-point DFT block128. A parallel output of the M-point DFT block 128 is coupled to aparallel input of the subcarrier demapping and equalization block 130. Aparallel output of the subcarrier demapping and equalization block 130is coupled to a parallel input of the N-point IDFT block 132. A paralleloutput of the N-point IDFT block 132 is coupled to a parallel input ofthe parallel to serial block 134. A serial output of the parallel toserial block 134 is coupled to a serial input of the detection block136.

In operation, for LTE uplink at the UE 101, a block of N modulationsymbols are received at the serial to parallel block 102 and are appliedas a parallel input to the N-point DFT block 104. The N-point DFT block104 performs a discrete Fourier transform on the modulation symbols andthen the output of the N-point DFT block 104 is applied to consecutiveinputs of the M-point IFFT block 108 (where M>N) via the subcarriermapping block 106. The output of the M-point IDFT block 108 is convertedto a serial signal by the parallel to serial block 110 and a cyclicprefix is applied to each block of the serial signal in the CP and PSblock 112. The signal is converted to an analogue signal and modulatedat radio frequency in the DAC and RF converter block 114 before beingtransmitted using the antenna 116 over the channel 118 to the antenna120 of the base station 121.

The signal received at the antenna 120 of the base station 121 isdemodulated and converted to a digital signal in the RF converter andADC block 122. The cyclic prefix is removed in the remove CP block 124.The signal is then converted to a parallel signal by the serial toparallel block 126 before the M-point DFT block 128 performs a discreteFourier transform of the signal. The output of the M-point DFT block 128is demapped and equalized by the subcarrier demapping and equalizationblock 130 before the N-point IDFT block 132 performs an inverse discreteFourier transform on the signal. The output of the N-point IDFT block132 is converted to a serial signal by parallel to serial block 134before being passed to the detection block 136 for detection.

Since typically for a RACH preamble, a DFT of size N=839 or 139 needs tobe taken (depending on the format of the preamble), the operationperformed in the N-point DFT block 104 is demanding in terms ofcomputational complexity and memory. One method for implementing a DFTwhere the size of the DFT is a prime number is the Bluestein algorithm(Leo I. Bluestein, “A linear filtering approach to the computation ofthe discrete Fourier transform,” Northeast Electronics Research andEngineering Meeting Record 10, 218-219 (1968)). In the Bluesteinalgorithm the DFT is re-expressed as a convolution which provides a wayto compute prime-size DFTs with a computational complexity of the orderO(N log N).

It is an aim of the present invention to reduce the computationalcomplexity required to perform a prime number DFT for use in processinga signal to be transmitted on a Random Access Channel.

SUMMARY

According to a first aspect of the invention there is provided a methodof processing a signal to be transmitted on a random access channel, thesignal being a chirp-like polyphase sequence, the method being arecursive method for determining a plurality of frequency components ofthe signal, comprising: determining a first frequency component of theplurality of frequency components; determining a component factor byaccessing a factor table for use in determining a second frequencycomponent of the plurality of frequency components; and determining thesecond frequency component using the determined first frequencycomponent and the determined component factor.

According to a second aspect of the invention there is providedapparatus for processing a signal to be transmitted on a random accesschannel using a recursive method for determining a plurality offrequency components of the signal, the signal being a chirp-likepolyphase sequence, the apparatus comprising: means for determining afirst frequency component of the plurality of frequency components;means for determining a component factor by accessing a factor table foruse in determining a second frequency component of the plurality offrequency components; and means for determining the second frequencycomponent using the determined first frequency component and thedetermined component factor.

According to a third aspect of the invention there is provided acomputer program product comprising computer readable instructions forexecution on a computer, the instructions being for processing a signalto be transmitted on a random access channel, the signal being achirp-like polyphase sequence, the instructions comprising instructionsfor executing a recursive method for determining a plurality offrequency components of the signal comprising the steps of: determininga first frequency component of the plurality of frequency components;determining a component factor by accessing a factor table for use indetermining a second frequency component of the plurality of frequencycomponents; and determining the second frequency component using thedetermined first frequency component and the determined componentfactor.

An efficient implementation of the DFT of a Zadoff-Chu sequence (or anyother chirp-like polyphase sequence) is provided without needing toperform a Fourier transform. The method uses a recursive relation withreduced complexity. The Zadoff-Chu sequence has been chosen to be usedfor RACH preambles in LTE wireless networks, so the ability to implementa Fourier transform with reduced complexity on Zadoff-Chu sequences isparticularly beneficial. However, it is noted that the method works withany signal that is a chirp-like polyphase sequence. The Zadoff Chusequence is just one example of a chirp-like polyphase sequence. Aswould be apparent to a skilled person, chirp-like polyphase sequenceshave ideal periodic autocorrelation functions. Details on chirp-likepolyphase sequences can be found in “Generalized Chirp-Like PolyphaseSequences with optimum Correlation Properties” by Branislav M. Popović,IEEE Transactions on Information Theory, vol. 38, No. 4, July 1992,pages 1406 to 1409. It is described in that reference that as well asZadoff-Chu sequences, Frank sequences and also Ipatov sequences arechirp-like polyphase sequences.

The complexity of implementing the Fourier Transform is reduced by usinga lookup table with a simple index computation. Such indexing requiresless processing power than performing a conventional DFT. The table maybe stored at the UE. Component factors in the table may be calculated bythe UE. Alternatively, the component factors stored in the table may becalculated by an entity other than the UE and passed to the UE forstorage thereon.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the present invention and to show how thesame may be put into effect, reference will now be made, by way ofexample, to the following drawings in which:

FIG. 1 is a schematic representation of a SC-FDMA communication system;and

FIG. 2 is a flow chart for a process of processing a signal according toa preferred embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Before describing a preferred embodiment of the invention, there isprovided a derivation of equations that are used in the preferredembodiment to facilitate the understanding of the invention.

As described above, the Zadoff-Chu sequence (for odd N_(ZC)) is definedby

${x(n)} = ^{{- j}\; \frac{\pi \; \mu \; {n{({n + 1})}}}{N_{ZC}}}$${x(n)} = ^{({- \frac{2j\; \pi \; \mu {\sum\limits_{i = 1}^{n}}}{N_{ZC}}})}$${x(n)} = {\prod\limits_{i = 1}^{n}^{({- \frac{2j\; \mu \; }{N_{ZC}}})}}$

where the fact that

${\sum\limits_{i = 1}^{n}i} = \frac{n\left( {n + 1} \right)}{2}$

has been used.

This can be rewritten as a recursive equation, such that:

${x(n)} = {{x\left( {n - 1} \right)}{^{- \frac{2j\; \pi \; \mu \; n}{N_{ZC}}}.}}$

Taking the Discrete Fourier transform of the above relation and usingthe DFT properties, one gets:

$\begin{matrix}{{{X(k)} = {{X\left( {k + \mu} \right)}^{- \frac{2j\; {\pi {({\mu + k})}}}{N_{ZC}}}}},} & (1)\end{matrix}$

where X(k) is the discrete Fourier transform of x(n).

Based on equation (1), recursively one can write using the shiftproperties of the DFT:

$\begin{matrix}{{X(k)} = {{X\left( {k + \mu} \right)}^{({- \frac{2j\; {\pi {({\mu + k})}}}{N_{zc}\;}})}}} \\{= {{X\left( {k + {2\mu}} \right)}^{({- \frac{2{{j\pi}{({\mu + k})}}}{N_{zc}}})}^{({- \frac{2j\; {\pi {({{2\mu} + k})}}}{N_{zc}}})}}} \\{\vdots} \\{= {{X\left( {k + {m\mspace{2mu} \mu}} \right)}{\prod\limits_{l = 1}^{m}^{({- \frac{2j\; {\mu {({{l\; \mu} + k})}}}{N_{zc}}})}}}} \\{= {{X\left( {k + {m\; \mu}} \right)}^{({- \frac{2j\; \pi {\sum\limits_{l = 1}^{m}{({k + {l\; \mu}})}}}{N_{zc}}})}}} \\{= {{X\left( {k + {m\; \mu}} \right)}^{- \frac{2j\; \pi \; {mk}}{N_{zc}}}^{{- j}\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{zc}}}}}\end{matrix}$

Let us introduce the following notation. Two integers a and b are saidto be congruent modulo n, if their difference a−b is an integer multipleof n. An equivalent definition is that both numbers have the sameremainder when divided by n. If this is the case, it is expressed as:

a=b mod n.

Let us choose m such that mμ=1 mod N_(ZC). m always exists since N_(ZC)and μ are relatively prime numbers (i.e. they share no common positivefactors, or divisors, except 1) by construction of the Zadoff-Chusequence. Then, from the periodicity property of the DFT:

X(k+mμ)=X(k+1),

one obtains the final result as:

$\begin{matrix}{{{X\left( {k + 1} \right)} = {{X(k)}^{\frac{2j\; \pi \; {mk}}{N_{ZC}}}^{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}}}{with}{{X(0)} = {\overset{N_{ZC} - 1}{\sum\limits_{n = 0}}{^{{- j}\; \pi \; \mu \; \frac{n{({n + 1})}}{N_{ZC}}}.}}}} & (2)\end{matrix}$

From equation (2), one can get an expression for X(k) as:

${X(k)} = {{X\left( {k - 1} \right)}^{\frac{2j\; \pi \; {m{({k - 1})}}}{N_{ZC}}}^{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}}$

By recursion one gets:

${X(k)} = {{X\left( {k - 2} \right)}^{\frac{2j\; \pi \; {m{({k - 2})}}}{N_{ZC}}}^{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}^{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}^{{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}\;}}$

which leads to:

${X(k)} = {{X(0)}{\prod\limits_{l = 0}^{k - 1}{^{\frac{2j\; \pi \; m\; l}{N_{ZC}}}^{{j\; \pi \; \mu \; \frac{m{({m + 1})}}{N_{ZC}}}\;}}}}$

and finally to the result that:

$\begin{matrix}{{X(k)} = {{X(0)}^{\frac{2j\; \pi \; {{mk}{({k - 1})}}}{N_{ZC}}}^{j\; \pi \; \mu \; \frac{k\; {m{({m + 1})}}}{N_{ZC}}}}} & (3)\end{matrix}$

In the case where N_(ZC) is even, the Zadoff-Chu sequence is given by:

${x(n)} = {^{{- j}\; \pi \; \mu \; \frac{n^{2}}{N_{ZC}}}.}$

One can show by inductive proof that:

${x(n)} = {{x\left( {n - 1} \right)}^{({- \frac{2j\; \pi \; \mu \; n}{N_{ZC}\;}})}{^{({- \frac{j\; \pi \; \mu}{N_{ZC}}})}.}}$

Similarly to the derivations above, one obtains:

$\begin{matrix}{{X(k)} = {{X\left( {k + \mu} \right)}^{- \frac{2j\; {\pi {({\mu + k})}}}{N_{ZC}}}^{- \frac{j\; \pi \; \mu}{N_{ZC}}}}} & (4)\end{matrix}$

which leads to:

${X(k)} = {{X\left( {k + {m\mspace{2mu} \mu}} \right)}{\prod\limits_{l = 1}^{m}{^{- \frac{2j\; {\pi {({{l\; \mu} + k})}}}{N_{ZC}}}^{- \frac{j\; \pi \; \mu}{N_{ZC}}}}}}$${X(k)} = {{X\left( {k + {m\; \mu}} \right)}^{- \frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{- j}\; \pi \; \mu \frac{\; {m{({m + 1})}}}{N_{ZC}}}^{- \frac{j\; \pi \; m\; \mu}{N_{ZC}}}}$

If m is such that mμ=1 mod N_(ZC) exists, then one can rewrite the aboveequation as:

$\begin{matrix}{{X(k)} = {{X\left( {k + 1} \right)}^{- \frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{- {j\pi}}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{- \frac{{j\pi}\; m\; \mu}{N_{ZC}}}}} & (5)\end{matrix}$

X(k) can be expressed as (if m is such that mμ=1 mod N_(ZC) exists):

${X(k)} = {{X(0)}{\prod\limits_{l = 0}^{k - 1}\; \left( {^{\frac{2j\; \pi \; {ml}}{N_{ZC}}}^{{j\pi}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{\frac{{j\pi}\; m\; \mu}{N_{ZC}}}} \right)}}$${X(k)} = {{X(0)}^{\frac{j\; \pi \; {{mk}{({k - 1})}}}{N_{ZC}}}^{{j\pi}\; \mu \frac{{km}{({m + 1})}}{N_{ZC}}}^{\frac{{j\pi}\; {km}\; \mu}{N_{ZC}}}}$

If m is such that mμ=1 mod N_(ZC) does not exist, one can find thesmallest integer β such that min{β|β<μ and mμ=β mod N_(ZC)} in order tominimize the delay, and Equation (3) becomes:

$\begin{matrix}{{X(k)} = {{X\left( {k + \beta} \right)}^{- \frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{- {j\pi}}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{- \frac{{j\pi}\; {\mu m}}{N_{ZC}}}}} & (6)\end{matrix}$

From the μth root of the Zadoff-Chu sequence, random access preambleswith zero correlation zones of length N_(CS)−1 are defined by cyclicshifts according to:

x_(μ, v) = x_(μ )((n + C_(v))mod N_(ZC))$x_{\mu} = ^{{- {j\pi\mu}}\frac{n{({n + 1})}}{N_{ZC}}}$

for 0≦n≦N_(ZC), where

$C_{v} = \left\{ \begin{matrix}{vN}_{CS} & {{v = 0},1,\ldots \mspace{14mu},{\left\lfloor {N_{ZC}/N_{CS}} \right\rfloor - 1},{N_{CS} \neq 0}} & {{for}\mspace{14mu} {unrestricted}\mspace{14mu} {sets}} \\0 & {N_{CS} = 0} & {{for}\mspace{14mu} {unrestricted}\mspace{14mu} {sets}} \\{{d_{start}\left\lfloor {v/n_{shift}^{RA}} \right\rfloor} + {\left( {v\; {mod}\; n_{shift}^{RA}} \right)N_{CS}}} & {{v = 0},1,\ldots \mspace{14mu},{{n_{shift}^{RA}n_{group}^{RA}} + n_{shift}^{- {RA}} - 1}} & {{for}\mspace{14mu} {restricted}\mspace{14mu} {sets}}\end{matrix} \right.$

and N_(CS) is signalled by high layers.

The DFT for a Zadoff-Chu sequence of odd length is given by Equation (2)above:

${x_{\mu}\left( {k + 1} \right)} = {{X_{\mu}(k)}^{\frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{j\pi}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}}$

The DFT of the vth cyclically shifted Zadoff-Chu sequence is given by:

${x_{\mu,v}(k)} = {{X_{\mu}(k)}{^{\frac{2j\; \pi \; C_{v}k}{N_{ZC}}}.}}$

Therefore by modifying the recursive equation (2) shown above, oneobtains:

$\begin{matrix}{{X_{\mu,v}\left( {k + 1} \right)} = {{X_{\mu,v}(k)}^{\frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{j\pi}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{\frac{2{j\pi}\; C_{v}k}{N_{ZC}}}}} & (7)\end{matrix}$

The exponential part of equation 7 for different values of k may bestored in a table at the user equipment, for use as component factors indetermining the frequency components of the signal, as described below.Obtaining the exponential part of equation 7 (i.e. a component factor)can then be easily implemented by indexing into the table of componentfactors which for ease of notation is restricted to a size of N_(ZC),which corresponds to 2π with a resolution of 2π/N_(ZC). In alternativeembodiments, a table of different resolution and length may be used.

In this way the exponential part of equation (7) (referred to herein asthe component factor) for different frequency components (k) iscalculated and stored in the table. Each frequency component of thesignal (X(k+1)) can be calculated using the previously calculatedfrequency component and a component factor obtained from the table. Inother words X(k+1)=X(k)F_(k+1), where F_(k+1) is the component factorfor the frequency component X(k+1) and is given by

${F_{k + 1} = {^{\frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{j\pi}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{\frac{2{j\pi}\; C_{v}}{N_{ZC}}}}},$

and the values of F_(k+1) can be determined by accessing the table in anindexed manner. The value of the exponent (divided by a factor of

$\left. \frac{2\pi \; j}{N_{ZC}} \right)$

is used as the index for accessing the table, as described in moredetail below. The values of F_(k) for the different frequency components(k) in the signal may be calculated at the user equipment 101 and storedin the table. Alternatively, the values of F_(k) for the differentfrequency components (k) in the signal may be calculated at an entityother than the user equipment 101 and stored in the table. The values ofF_(k) for the different frequency components (k) in the signal may becalculated before they are needed and stored in the table before theyare needed. In this way, when the factors F_(k) are needed they justneed to be looked up from the table rather than calculated. The table isstored in memory of the user equipment.

A method of carrying out the invention according to a preferredembodiment is now described with reference to the flow chart of FIG. 2.The method is carried out to implement a DFT, for use in a method ofprocessing signals for transmission on a RACH, e.g. where the signalsare RACH preambles. For example, the method may be implemented in theN-point DFT block 104 as shown in the system of FIG. 1.

In step S202 the frequency component X(0) is determined. X(0) may bedetermined by loading the frequency component from a store.Alternatively, X(0) may be calculated from the signal. Once thefrequency component X(0) of the signal has been determined, the otherfrequency components in the signal can be calculated in a recursivemanner using equation (7) above.

In order to begin the recursive method, in step S204, a counter i is setto 1 initially. Then in step S206 the component factor F_(i) for the ithfrequency component of the signal is looked up by indexing the table.Therefore on the first run through of the recursive method the componentfactor F₁ is obtained from the table. In step S208 the ith frequencycomponent (X(i)) is determined using the previously determined frequencycomponent (X(i−1)) and the component factor for the ith frequencycomponent obtained in step S206. In the first run through of therecursive method the frequency component X(1) is determined bymultiplying X(0) with F₁. In this sense, the component factors F_(i)stored in the table are multiplying factors. Alternatively, thecomponent factors F_(i) stored in the table may be used to obtain theith frequency component in other ways than by multiplication with apreviously determined frequency component. The component factorsobtained from the table may be used in conjunction with a previouslydetermined frequency component in any way, as would be apparent to theskilled person, leading to a determination of the ith frequencycomponent of the signal.

In the embodiments described above the component factors F_(k) arestored in the table. This is a much simpler operation than calculatingthe DFT for each component factor.

In step S210 the counter i is incremented by 1 and in step S212 it isdetermined whether the counter i is greater than or equal to the lengthof the Zadoff-Chu sequence N_(ZC). If the counter is greater than orequal to N_(ZC) then all of the frequency components of the signal havebeen determined and the process ends in step S214. However, if thecounter i is less than N_(ZC) then the method passes back to step S206and the next frequency component of the signal is determined. Theprocess continues until all of the frequency components of the signalhave been determined.

In this way, a running table index is obtained which is initialised to

${\frac{\mu \; {m\left( {m + 1} \right)}}{2} + C_{v}} = {{I\; {mod}\; N_{ZC}\mspace{14mu} {for}\mspace{14mu} k} = 0.}$

Using this index when accessing the table will return the value of thecomponent factor F₁, given by

$F_{1} = ^{\frac{2j\; \pi}{N_{ZC}}{({\frac{\mu \; {m{({m + 1})}}}{2} + C_{v}})}}$

(see the equation above for F_(k+1)). This component factor is thenmultiplied with X(0) to give X(1). The next pass in the recursionrequires I to be updated by m as:

m=γ mod N_(ZC)

I _(i) +γ=I _(i+1) mod N _(ZC)

where I_(i) is the index at iteration i. Note that the modulo operationdoes not need a divide since (I+γ) can never exceed 2N_(ZC).

Pseudo code which may be used to implement the above described methodwill now be described. The pseudo code may be implemented in a computerprogram product for execution on a computer or other suitable hardwarefor carrying out the method as described above. Alternatively, themethod may be carried out in hardware, rather than in software, as wouldbe apparent to a skilled person.

In the following the notation a=b mod n is equivalent to b=mod(a,n).

The pseudo code may be written as follows:

  load X(0) from a pre computed table or compute X(0) Compute γ =mod(m,Nzc) initialise I to mod(μm(m+1)/2 +Cv,Nzc) for 1=1to Nzc−1  e =load exp from table with index I  X(i)=X(i−1)*e  I=I+γ  If I>=NzcI=I−Nzc endfor

It will be apparent to a person skilled in the art that using the methoddescribed above, as provided for by the pseudo code above, the frequencycomponents X(i) can be determined in a recursive manner, thus requiringless computing power and complexity than performing a conventionalFourier transform to determine the frequency components X(i).

In an alternative embodiment, the computational complexity may bereduced even further. If the signal may be multiplied by a complexconstant we can ensure that the frequency component X(0)=1. In this waythe first frequency component is set to 1 so it does not need to becomputed. Multiplying the signal by a complex constant is equivalent toa scaling introduced in the communication channel 118, which changesneither the received timing nor RACH detection probability nor falsealarm probability as determined by the base station 121. Thereforemultiplying the signal by a complex constant does not detrimentallyaffect the use of the signal as a RACH preamble.

Where the signal is multiplied by a complex constant to ensure that thefrequency component X(0) is equal to 1, the multiplication of thecomponent factor obtained from the factor table and the previouslydetermined frequency component in the recursive method described abovemay be avoided altogether. In this case the algorithm may be modified tohave the following pseudo code.

  Set X(0) = 1 Compute γ= mod(m,Nzc) Initialise I to mod(μm(m+1)/2+Cv,Nzc) J=0 for i=1 to Nzc−1  J=J+I  If J>=Nzc, J=J−Nzc  e = load expfrom table with index J  X(i)=e  I=I+γ  If I>=Nzc I=I−Nzc endfor

In this alternative embodiment, it can be seen from equation (7) thatwith X(0) equal to 1, all of the frequency components will equal e^(d)^(i) where d_(i) is different for each of the frequency components X(i).Since e^(a)e^(b)=e^(a+b), each frequency component can be computed byadding up the indices for all previous frequency components and usingthe sum of the indices as the index for accessing the table. The pseudocode given above for this alternative embodiment implements this byloading the component factors using the index J where J is the sum ofall previously determined indices I.

As would be apparent to the skilled person, a similar implementationapproach could be used for the case where N_(ZC) is even based onEquations 5 and 6.

There have been described above methods of implementing a Fouriertransform for use in the signal processing of RACH preambles using theZadoff-Chu sequence. The methods described above do not use a dedicatedFourier transform algorithm for calculating the frequency components ofthe signal. This results in reduced complexity and reduced memoryrequirements. The implementation of the DFT is simplified by using atable-lookup with index computation.

While the specific description is directed towards signal processing ofsignals using the Zadoff-Chu sequence, it would be apparent to a skilledperson that the method could also be applied to any other chirp-likepolyphase sequence.

While this invention has been particularly shown and described withreference to preferred embodiments, it will be understood to thoseskilled in the art that various changes in form and detail may be madewithout departing from the scope of the invention as defined by theappendant claims.

1. A method of processing a signal, the signal being a chirp-likepolyphase sequence, the method being a recursive method for determininga plurality of frequency components of the signal, comprising:determining a first frequency component of the plurality of frequencycomponents; determining a component factor by accessing a factor tablefor use in determining a second frequency component of the plurality offrequency components; and determining the second frequency componentusing the determined first frequency component and the determinedcomponent factor.
 2. The method of claim 1 wherein there is at least onefurther frequency component of the signal and the method furthercomprises for each of the further frequency components: determining arespective further component factor by accessing the factor table foruse in determining the further frequency component; and determining thefurther frequency component using a previously determined frequencycomponent and the determined further component factor.
 3. The method ofclaim 2 wherein the previously determined frequency component is thefrequency component determined most recently prior to determining eachrespective further frequency component.
 4. The method of claim 2 whereinthe previously determined frequency component is the first frequencycomponent.
 5. The method of claim 1 further comprising setting the firstfrequency component to 1 so it does not need to be computed
 6. Themethod of claim 1 wherein the factor table stores a plurality ofcomponent factors in an indexed manner for use in determining theplurality of frequency components of the signal, and wherein the step ofobtaining the component factor comprises indexing the factor table withan index corresponding to the component factor.
 7. The method of claim 6further comprising calculating the plurality of component factors andstoring the calculated component factors in the factor table.
 8. Themethod of claim 1 wherein the step of determining the first frequencycomponent comprises loading the first frequency component from a store.9. The method of claim 1 wherein the step of determining the firstfrequency component comprises calculating the first frequency componentfrom the signal.
 10. The method of claim 1 wherein the component factoris a multiplication factor, and wherein the step of determining thesecond frequency component comprises multiplying the first frequencycomponent by the component factor.
 11. The method of claim 1 wherein thesignal is a Zadoff-Chu sequence in which the signal, x, at eachposition, n, of each root, μ, of the Zadoff-Chu sequence is given by${{x_{\mu}(n)} = ^{{- {j\pi\mu}}\frac{n{({n + 1})}}{N_{ZC}}}},$where N_(ZC) is the length of the Zadoff-Chu sequence.
 12. The method ofclaim 11 wherein the length of the Zadoff-Chu sequence is a primenumber.
 13. The method of claim 12 wherein the length of the Zadoff-Chusequence is 139 or
 839. 14. The method of claim 11 wherein the (k+1)thfrequency component, X(k+1), of the signal is determined using the kthfrequency component, X(k), of the signal and the (k+1)th componentfactor, F_(k+1), according to the formula:X(k+1)=X(k)F _(k+1), where${F_{k + 1} = {^{\frac{2j\; \pi \; {mk}}{N_{ZC}}}^{{j\pi}\; \mu \frac{m{({m + 1})}}{N_{ZC}}}^{\frac{2{j\pi}\; C_{v}}{N_{ZC}}}}},$and where m is an integer chosen such that mμ=1 mod N_(ZC), and C_(v) isan integer defined by the equation x_(μ,v)=x_(μ)((n+C_(v))mod N_(ZC))where for the μth root of the Zadoff-Chu sequence, x_(μ,v) is acyclically shifted version of x_(μ) wherein x_(μ,v) and x_(μ) have zerocorrelation zones of length N_(CS)−1.
 15. The method of claim 11 furthercomprising: setting an increment variable, γ; and setting an indexvariable I, wherein said step of determining a component factorcomprises loading the component factor from the factor table using theindex variable I, the method further comprising: incrementing the indexvariable I by the increment variable γ for determining a subsequentcomponent factor by loading the subsequent component factor from thefactor table using the incremented index variable.
 16. The method ofclaim 11 when dependent upon claim 5 further comprising: setting anincrement variable, γ; setting an index variable I; setting a loadingvariable J such that J=I initially, wherein said step of determining acomponent factor comprises loading the component factor from the factortable using the loading variable J, and said step of determining thesecond frequency component comprises setting the second frequencycomponent to be the determined component factor, the method furthercomprising: incrementing the index variable I by the increment variableγ; and incrementing the loading variable J by the index variable I fordetermining a subsequent component factor by loading the subsequentcomponent factor from the factor table using the incremented loadingvariable.
 17. The method of claim 15 wherein the increment variable γ isset such that γ=mod(m, N_(ZC)), where m is an integer chosen such thatmμ=1 mod N_(ZC), and the index variable I is set such that$I = {{{mod}\left( {{{\mu \; m\frac{\left( {m + 1} \right)}{2}} + C_{v}},N_{ZC}} \right)}.}$18. The method of claim 1 further comprising applying the determinedfrequency components to consecutive inputs of an Inverse Fast FourierTransform block.
 19. The method of claim 1 further comprising:separating the signal into blocks for transmission; and inserting acyclic prefix into each block.
 20. The method of claim 1 furthercomprising transmitting the signal on the random access channel.
 21. Themethod of claim 1 wherein the signal is a random access channelpreamble.
 22. The method of claim 1 wherein the signal is transmitted inthe uplink on the random access channel.
 23. Apparatus for processing asignal to be transmitted on a random access channel using a recursivemethod for determining a plurality of frequency components of thesignal, the signal being a chirp-like polyphase sequence, the apparatuscomprising: means for determining a first frequency component of theplurality of frequency components; means for determining a componentfactor by accessing a factor table for use in determining a secondfrequency component of the plurality of frequency components; and meansfor determining the second frequency component using the determinedfirst frequency component and the determined component factor.
 24. Theapparatus of claim 23 further comprising means for storing the factortable.
 25. The apparatus of claim 23 further comprising an Inverse FastFourier Transform block, the apparatus being configured to apply thedetermined frequency components of the signal to consecutive inputs ofthe Inverse Fast Fourier Transform block.
 26. The apparatus of claim 23further comprising: means for separating the signal into blocks fortransmission; and means for inserting a cyclic prefix into each block.27. The apparatus of claim 23 further comprising a transmitter fortransmitting the signal.
 28. A computer program product comprisingcomputer readable instructions stored on a non-transitory computerreadable medium for execution on a computer, the instructions being forprocessing a signal to be transmitted on a random access channel, thesignal being a chirp-like polyphase sequence, the instructionscomprising instructions for executing a recursive method for determininga plurality of frequency components of the signal comprising the stepsof: determining a first frequency component of the plurality offrequency components; determining a component factor by accessing afactor table for use in determining a second frequency component of theplurality of frequency components; and determining the second frequencycomponent using the determined first frequency component and thedetermined component factor.